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The study of interaction between different degrees of freedom in solids is offundamental importance to understand the functionalities of materials. Onestriking example of such interaction is the intertwined coupling or competitionbetween superconductivity (SC), charge density wave (CDW), pseudogap state(PG), and other exotic phases in cuprate superconductors. Recent emergence ofnonlinear Terahertz (THz) third harmonic generation (THG) spectroscopy providesa powerful tool for exploring the collective (Higgs) modes of superconductivityorder parameters, and its interaction with intertwined/competing phases. Inthis study, we report on nonlinear THz THG spectroscopy of theYBa$_2$Cu$_3$O$_{6+x}$ (YBCO) thin films with different doping. We identify acharacteristic temperature $T_{THG}$, below which third order suscepetility$\chi^{(3)}$ emerges. Notably, the $T_{THG}$ is coincident with the crossovertemperature $T^*$ of pseudogap in a wide range doping of phase diagram. Uponentering the superconducting state, THG increases sharply but exhibits anabnormal dip feature near $T_c$ which is more clearly seen in optimally dopedsample. Strikingly, we observe a beating structure directly in the measuredreal time waveform of THG signal. Fourier transformation of the time domainwaveform gives two separate modes below and above original THG frequency. Theobservation strongly indicates that an additional mode, presumably Higgs mode,appears at $T_c$ and couples to the mode already developed below $T^*$. Thestrong coupling effect offers new insight into the interplay betweensuperconductivity and pseudogap. The result unambiguously suggests that thepseudogap phase is not a precursor of superconductivity but represents adistinct order.

Revealing strong coupling of collective modes between superconductivity  and pseudogap in cuprate superconductor by terahertz third harmonic  generation